Question: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $x = \dfrac{2q(3q + 4)}{9} \div \dfrac{6q + 8}{-5} $
Dividing by an expression is the same as multiplying by its inverse. $x = \dfrac{2q(3q + 4)}{9} \times \dfrac{-5}{6q + 8} $ When multiplying fractions, we multiply the numerators and the denominators. $x = \dfrac{ 2q(3q + 4) \times -5 } { 9 \times (6q + 8) } $ $ x = \dfrac {-5 \times 2q(3q + 4)} {9 \times 2(3q + 4)} $ $ x = \dfrac{-10q(3q + 4)}{18(3q + 4)} $ We can cancel the $3q + 4$ so long as $3q + 4 \neq 0$ Therefore $q \neq -\dfrac{4}{3}$ $x = \dfrac{-10q \cancel{(3q + 4})}{18 \cancel{(3q + 4)}} = -\dfrac{10q}{18} = -\dfrac{5q}{9} $